The **tidal force** or **tide-generating force** is a gravitational effect that stretches a body along the line towards and away from the center of mass of another body due to spatial variations in strength in gravitational field from the other body. It is responsible for the tides and related phenomena, including solid-earth tides, tidal locking, breaking apart of celestial bodies and formation of ring systems within the Roche limit, and in extreme cases, spaghettification of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearer side is attracted more strongly than the farther side. The difference is positive in the near side and negative in the far side, which causes a body to get stretched. Thus, the tidal force is also known as the differential force, residual force, or secondary effect of the gravitational field.

- Explanation
- Size and distance
- Sun, Earth, and Moon
- Effects
- Formulation
- See also
- References
- External links

In celestial mechanics, the expression *tidal force* can refer to a situation in which a body or material (for example, tidal water) is mainly under the gravitational influence of a second body (for example, the Earth), but is also perturbed by the gravitational effects of a third body (for example, the Moon). The perturbing force is sometimes in such cases called a tidal force^{ [2] } (for example, the perturbing force on the Moon): it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first.^{ [3] }

Tidal forces have also been shown to be fundamentally related to gravitational waves.^{ [4] }

When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 4 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2). These so-called *tidal forces* cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart.^{ [5] } The Roche limit is the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of the object for one another.^{ [6] } These strains would not occur if the gravitational field were uniform, because a uniform field only causes the entire body to accelerate together in the same direction and at the same rate.

The relationship of an astronomical body's size, to its distance from another body, strongly influences the magnitude of tidal force.^{ [7] } The tidal force acting on an astronomical body, such as the Earth, is directly proportional to the diameter of that astronomical body and inversely proportional to the cube of the distance from another body producing a gravitational attraction, such as the Moon or the Sun. Tidal action on bath tubs, swimming pools, lakes, and other small bodies of water is negligible.^{ [8] }

Figure 3 is a graph showing how gravitational force declines with distance. In this graph, the attractive force decreases in proportion to the square of the distance (*Y* = 1/*X*^{2}), while the slope (*Y*′ = −2/*X*^{3}) is inversely proportional to the cube of the distance.

The tidal force corresponds to the difference in Y between two points on the graph, with one point on the near side of the body, and the other point on the far side. The tidal force becomes larger, when the two points are either farther apart, or when they are more to the left on the graph, meaning closer to the attracting body.

For example, even though the Sun has a stronger overall gravitational pull on Earth, the Moon creates a larger tidal bulge because the Moon is closer. This difference is due to the way gravity weakens with distance: the Moon's closer proximity creates a steeper decline in its gravitational pull as you move across Earth (compared to the Sun's very gradual decline from its vast distance). This steeper gradient in the Moon's pull results in a larger difference in force between the near and far sides of Earth, which is what creates the bigger tidal bulge.

Gravitational attraction is inversely proportional to the square of the distance from the source. The attraction will be stronger on the side of a body facing the source, and weaker on the side away from the source. The tidal force is proportional to the difference.^{ [8] }

The Earth is 81 times more massive than the Moon but has roughly 4 times its radius. As a result, at the same distance, the tidal force of the Earth at the surface of the Moon is about 20 times stronger than that of the Moon at the Earth's surface.^{ [9] }

Gravitational body causing tidal force | Body subjected to tidal force | Tidal acceleration | |||
---|---|---|---|---|---|

Body | Mass () | Body | Radius () | Distance () | |

Sun | 1.99×10^{30} kg | Earth | 6.37×10^{6} m | 1.50×10^{11} m | 5.05×10^{−7} m⋅s^{−2} |

Moon | 7.34×10^{22} kg | Earth | 6.37×10^{6} m | 3.84×10^{8} m | 1.10×10^{−6} m⋅s^{−2} |

Earth | 5.97×10^{24} kg | Moon | 1.74×10^{6} m | 3.84×10^{8} m | 2.44×10^{−5} m⋅s^{−2} |

G is the gravitational constant = 6.674×10^{−11} m^{3}⋅kg^{−1}⋅s^{−2}^{ [10] } |

In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid with two bulges, pointing towards and away from the other body. Larger objects distort into an ovoid, and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. The Earth and Moon orbit about their common center of mass or barycenter, and their gravitational attraction provides the centripetal force necessary to maintain this motion. To an observer on the Earth, very close to this barycenter, the situation is one of the Earth as body 1 acted upon by the gravity of the Moon as body 2. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon.^{ [12] }

When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. In the case for the Earth, and Earth's Moon, the loss of rotational kinetic energy results in a gain of about 2 milliseconds per century. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io. Stresses caused by tidal forces also cause a regular monthly pattern of moonquakes on Earth's Moon.^{ [7] }

Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that variations in tidal forces correlate with cool periods in the global temperature record at 6- to 10-year intervals,^{ [13] } and that harmonic beat variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date.^{ [14] }

Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes, where they are responsible for the "spaghettification" of infalling matter. Tidal forces create the oceanic tide of Earth's oceans, where the attracting bodies are the Moon and, to a lesser extent, the Sun. Tidal forces are also responsible for tidal locking, tidal acceleration, and tidal heating. Tides may also induce seismicity.

By generating conducting fluids within the interior of the Earth, tidal forces also affect the Earth's magnetic field.^{ [15] }

For a given (externally generated) gravitational field, the **tidal acceleration** at a point with respect to a body is obtained by vector subtraction of the gravitational acceleration at the center of the body (due to the given externally generated field) from the gravitational acceleration (due to the same field) at the given point. Correspondingly, the term *tidal force* is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant. (In other words, the comparison is with the conditions at the given point as they would be if there were no externally generated field acting unequally at the given point and at the center of the reference body. The externally generated field is usually that produced by a perturbing third body, often the Sun or the Moon in the frequent example-cases of points on or above the Earth's surface in a geocentric reference frame.)

Tidal acceleration does not require rotation or orbiting bodies; for example, the body may be freefalling in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.

By Newton's law of universal gravitation and laws of motion, a body of mass *m* at distance *R* from the center of a sphere of mass *M* feels a force ,

equivalent to an acceleration ,

where is a unit vector pointing from the body *M* to the body *m* (here, acceleration from *m* towards *M* has negative sign).

Consider now the acceleration due to the sphere of mass *M* experienced by a particle in the vicinity of the body of mass *m*. With *R* as the distance from the center of *M* to the center of *m*, let ∆*r* be the (relatively small) distance of the particle from the center of the body of mass *m*. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass *M*. If the body of mass *m* is itself a sphere of radius ∆*r*, then the new particle considered may be located on its surface, at a distance (*R* ± *∆r*) from the centre of the sphere of mass *M*, and *∆r* may be taken as positive where the particle's distance from *M* is greater than *R*. Leaving aside whatever gravitational acceleration may be experienced by the particle towards *m* on account of *m*'s own mass, we have the acceleration on the particle due to gravitational force towards *M* as:

Pulling out the *R*^{2} term from the denominator gives:

The Maclaurin series of is which gives a series expansion of:

The first term is the gravitational acceleration due to *M* at the center of the reference body , i.e., at the point where is zero. This term does not affect the observed acceleration of particles on the surface of *m* because with respect to *M*, *m* (and everything on its surface) is in free fall. When the force on the far particle is subtracted from the force on the near particle, this first term cancels, as do all other even-order terms. The remaining (residual) terms represent the difference mentioned above and are tidal force (acceleration) terms. When ∆*r* is small compared to *R*, the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration for the distances ∆*r* considered, along the axis joining the centers of *m* and *M*:

When calculated in this way for the case where ∆*r* is a distance along the axis joining the centers of *m* and *M*, is directed outwards from to the center of *m* (where ∆*r* is zero).

Tidal accelerations can also be calculated away from the axis connecting the bodies *m* and *M*, requiring a vector calculation. In the plane perpendicular to that axis, the tidal acceleration is directed inwards (towards the center where ∆*r* is zero), and its magnitude is in linear approximation as in Figure 4.

The tidal accelerations at the surfaces of planets in the Solar System are generally very small. For example, the lunar tidal acceleration at the Earth's surface along the Moon–Earth axis is about 1.1×10^{−7} *g*, while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about 0.52×10^{−7} *g*, where *g* is the gravitational acceleration at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon.^{ [17] } The solar tidal acceleration at the Earth's surface was first given by Newton in the * Principia *.^{ [18] }

In physics, a **force** is an influence that can cause an object to change its velocity, i.e., to accelerate, meaning a change in speed or direction, unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol **F**.

**Mass** is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a body, until the discovery of the atom and particle physics. It was found that different atoms and different elementary particles, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple definitions which are conceptually distinct, but physically equivalent. Mass can be experimentally defined as a measure of the body's inertia, meaning the resistance to acceleration when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies.

In celestial mechanics, an **orbit** is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

In physics, **potential energy** is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. The term *potential energy* was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of *potentiality*.

In celestial mechanics, **escape velocity** or **escape speed** is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming:

**Newton's laws of motion** are three laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for **Newtonian mechanics**, can be paraphrased as follows:

- A body remains at rest, or in motion at a constant speed in a straight line, except insofar as it is acted upon by a force.
- At any instant of time, the net force on a body is equal to the body's acceleration multiplied by its mass or, equivalently, the rate at which the body's momentum is changing with time.
- If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.

In Newtonian physics, **free fall** is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

In physics, a **gravitational field** or **gravitational acceleration field** is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as the *gravitational force field* exerted on another massive body. It has dimension of acceleration (L/T^{2}) and it is measured in units of newtons per kilogram (N/kg) or, equivalently, in meters per second squared (m/s^{2}).

A **trajectory** or **flight path** is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously.

**Newton's law of universal gravitation** says that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Separated objects attract and are attracted as if all their mass were concentrated at their centers. The publication of the law has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.

In classical mechanics, the **gravitational potential** is a scalar field associating with each point in space the work per unit mass that would be needed to move an object to that point from a fixed reference point. It is analogous to the electric potential with mass playing the role of charge. The reference point, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance.

In physical theories, a **test particle**, or **test charge**, is an idealized model of an object whose physical properties are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behavior of the rest of the system. The concept of a test particle often simplifies problems, and can provide a good approximation for physical phenomena. In addition to its uses in the simplification of the dynamics of a system in particular limits, it is also used as a diagnostic in computer simulations of physical processes.

In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the **tidal tensor** represents

*tidal accelerations*of a cloud of test particles,*tidal stresses*in a small object immersed in an ambient gravitational field.

In physics, a **force field** is a vector field corresponding with a non-contact force acting on a particle at various positions in space. Specifically, a force field is a vector field , where is the force that a particle would feel if it were at the point .

In physics, **gravitational acceleration** is the acceleration of an object in free fall within a vacuum. This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies accelerate in vacuum at the same rate, regardless of the masses or compositions of the bodies; the measurement and analysis of these rates is known as gravimetry.

The **gravity of Earth**, denoted by **g**, is the net acceleration that is imparted to objects due to the combined effect of gravitation and the centrifugal force . It is a vector quantity, whose direction coincides with a plumb bob and strength or magnitude is given by the norm .

A set of **equations** describing the trajectories of objects subject to a constant gravitational force under normal Earth-bound conditions. Assuming constant acceleration *g* due to Earth’s gravity, Newton's law of universal gravitation simplifies to *F* = *mg*, where *F* is the force exerted on a mass *m* by the Earth’s gravitational field of strength *g*. Assuming constant *g* is reasonable for objects falling to Earth over the relatively short vertical distances of our everyday experience, but is not valid for greater distances involved in calculating more distant effects, such as spacecraft trajectories.

In physics and astronomy, an ** N-body simulation** is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity.

In relativity theory, **proper acceleration** is the physical acceleration experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Gravitation therefore does not cause proper acceleration, because the same gravity acts equally on the inertial observer. As a consequence, all inertial observers always have a proper acceleration of zero.

**Orbit modeling** is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects. Directly modeling an orbit can push the limits of machine precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit in order to achieve better accuracy.

- ↑ "Hubble Views a Cosmic Interaction".
*nasa.gov*. NASA. February 11, 2022. Retrieved 2022-07-09. - ↑ "On the tidal force", I. N. Avsiuk, in "Soviet Astronomy Letters", vol. 3 (1977), pp. 96–99.
- ↑ See p. 509 in "Astronomy: a physical perspective", M. L. Kutner (2003).
- ↑ arXiv, Emerging Technology from the (2019-12-14). "Tidal forces carry the mathematical signature of gravitational waves".
*MIT Technology Review*. Retrieved 2023-11-12. - ↑ R Penrose (1999).
*The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics*. Oxford University Press. p. 264. ISBN 978-0-19-286198-6.tidal force.

- ↑ Thérèse Encrenaz; J -P Bibring; M Blanc (2003).
*The Solar System*. Springer. p. 16. ISBN 978-3-540-00241-3. - 1 2 "The Tidal Force | Neil deGrasse Tyson".
*www.haydenplanetarium.org*. Retrieved 2016-10-10. - 1 2 Sawicki, Mikolaj (1999). "Myths about gravity and tides".
*The Physics Teacher*.**37**(7): 438–441. Bibcode:1999PhTea..37..438S. CiteSeerX 10.1.1.695.8981 . doi:10.1119/1.880345. ISSN 0031-921X. - ↑ Schutz, Bernard (2003).
*Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity*(illustrated ed.). Cambridge University Press. p. 45. ISBN 978-0-521-45506-0. Extract of page 45 - ↑ "2018 CODATA Value: Newtonian constant of gravitation".
*The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 May 2019. Retrieved 2019-05-20. - ↑ R. S. MacKay; J. D. Meiss (1987).
*Hamiltonian Dynamical Systems: A Reprint Selection*. CRC Press. p. 36. ISBN 978-0-85274-205-1. - ↑ Rollin A Harris (1920).
*The Encyclopedia Americana: A Library of Universal Knowledge*. Vol. 26. Encyclopedia Americana Corp. pp. 611–617. - ↑ Keeling, C. D.; Whorf, T. P. (5 August 1997). "Possible forcing of global temperature by the oceanic tides".
*Proceedings of the National Academy of Sciences*.**94**(16): 8321–8328. Bibcode:1997PNAS...94.8321K. doi: 10.1073/pnas.94.16.8321 . PMC 33744 . PMID 11607740. - ↑ Munk, Walter; Dzieciuch, Matthew; Jayne, Steven (February 2002). "Millennial Climate Variability: Is There a Tidal Connection?".
*Journal of Climate*.**15**(4): 370–385. Bibcode:2002JCli...15..370M. doi: 10.1175/1520-0442(2002)015<0370:MCVITA>2.0.CO;2 . - ↑ "Hungry for Power in Space".
*New Scientist*.**123**: 52. 23 September 1989. Retrieved 14 March 2016. - ↑ "Inseparable galactic twins".
*ESA/Hubble Picture of the Week*. Retrieved 12 July 2013. - ↑ The Admiralty (1987).
*Admiralty manual of navigation*. Vol. 1. The Stationery Office. p. 277. ISBN 978-0-11-772880-6., Chapter 11, p. 277 - ↑ Newton, Isaac (1729).
*The mathematical principles of natural philosophy*. Vol. 2. p. 307. ISBN 978-0-11-772880-6., Book 3, Proposition 36, Page 307 Newton put the force to depress the sea at places 90 degrees distant from the Sun at "1 to 38604600" (in terms of*g*), and wrote that the force to raise the sea along the Sun-Earth axis is "twice as great" (i.e., 2 to 38604600) which comes to about 0.52 × 10^{−7}*g*as expressed in the text.

- Analysis and Prediction of Tides: GeoTide
- Gravitational Tides by J. Christopher Mihos of Case Western Reserve University
- Audio: Cain/Gay – Astronomy Cast Tidal Forces – July 2007.
- Gray, Meghan; Merrifield, Michael. "Tidal Forces".
*Sixty Symbols*. Brady Haran for the University of Nottingham. - Pau Amaro Seoane. "Stellar collisions: Tidal disruption of a star by a massive black hole" . Retrieved 2018-12-28.
- Myths about Gravity and Tides by Mikolaj Sawicki of John A. Logan College and the University of Colorado.
- Tidal Misconceptions by Donald E. Simanek

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